Global behavior of epidemics models in age-structured populations(Mathematical Topics in Biology)

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Global behavior

of epidemics

models

in

age-structured

populations

Michel

LANGLAIS

URA CNRS

226,

CeReMaB

UFR MI2S,

University

of

Bordeaux

\Pi ,

33076

Bordeaux

Cedex,

France.

e-mail: [email protected]

Thecouplingbetween time periodic variations and the age-structureofa single species

population is investigated through a mathematical model also containing a spatial

struc-ture. Using simplifying assumptions we exhibit a threshold parameter yielding the

exis-tence and stability of non trivial stationary or periodic states. Next the propagation ofa

mild disease within this population is analyzed. More precisely, we look for sufficient

con-ditions giving, for the S.I.S. model with vertical transmission, the existence and stability

of a non trivial periodic endemic state under either a periodic contact rate or a periodic

supply of infected individuals. Stationary solutions are investigated as well.

1 INTRODUCTION

Let $u(x, t, a)=u\geq 0$ be the density of individuals in a single species population having age $a>0$ at time $t>0$ and location $x$ in some domain $\Omega$ in $1R^{d},$$d=1,2$ or 3 ; the usual

time-space distribution i.e. the total population is thus given by

$P(x,t)= \int_{0}^{\infty}u(x, t, a)da$ (1.1)

The dynamics of the population is run by the classical balance law of the Lotka and

Sharpe form

net growth-rate $=diffusion- death+supply$ ;

see Gurtin (1973), Hoppensteadt (1974) and the books of Webb (1985), Busenberg and

Cooke (1993). Assumingthe flux ofpopulation liesalongthe linesof (spatially) decreasing

densities this reads

$\partial_{t}u+\partial_{a}u=k\Delta_{x}u-\mu(P(x, t),$_$x,t,$$a$)$u+f(x,t, a)$

.

(1.2)

Herein, $\mu=\mu(p, x,t, a)\geq 0$ is the death rate at age $a>0$, time $t>0$ and location $x$

when the size of the total population is $p$, and $f=f(x, t, a)\geq 0$ the external supply

of individuals ; $k$ is the diffusion coefficient and $A_{x}$ the Laplace operator in the spatial

variables.

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$u(x, t, 0)= \int_{0}^{\infty}\beta(P(x, t),$$x,$$t,$ $a$)$u(x,t, a)da$, (1.3)

$\beta=\beta(p, x, t, a)\geq 0$ beingthe fertility function age $a>0$, time$t>0$ and location$x$ when

the size of the total population is $p$

.

One assumes a no flux boundary condition,

$\partial_{\eta}u(x,t, a)=0$ on the boundary of $\Omega$ (1.4)

The model equations for the epidemic problemare of the Kermack and Mac Kendrick

form, assuming the disease does affect neither the flux of population nor the birth and

death processes. The epidemic classes are composed of susceptible, infected and removed

individuals, denoted by $s,$$i$ and _$r$ respectively. Thus, setting $D=\partial_{t}+\partial_{a}-k\triangle_{x}$, one has

$\{\begin{array}{l}Ds+\mu(P(x,t),x,t,a)s=-\gamma(t,a)\varphi(i)s+\delta(a)i+\rho(a)r+f_{s}1Di+\mu(P(x,t),x,t,a)i=+\gamma(t,a)\varphi(i)s-\delta(a)i-\sigma(a)i+f_{i}Dr+\mu(P(x,t),x,t,a)r=+\sigma(a)i-\rho(a)r+f_{r}\end{array}$ (1.5)

where $\sigma$ (resp. $\delta$) is the age specific recovery rate with immunization (resp. without

immunization) and $\rho$ the rate at which this immunization is lost. The force of infection

$\gamma(t, a)\varphi(i)$ will take either mass action forms

$\gamma(t, a)\varphi(i)(x,t,a)=\{\begin{array}{l}\gamma_{0}(t,a)i(x,t,a)\gamma_{1}(t,a)\int_{0}^{\infty}i(x,t,a)da\end{array}$ $intercohortmodelintracohortmodel.$ ’

In the first case this says that a susceptible can get the disease only from an infected of

the same age while on the second case he may catch it from any infected individuals; $\gamma$

is the contact rate.

We do not consider here the general force of infection term of Busenberg and Cooke

(1993). The vertical transmission to offsprings is

$\{\begin{array}{l}s(x,t,0)=u(x,t,0)-i(x,t,0)-r(x,t,0)i(x,t,0)=\epsilon_{i}\int_{0}^{\infty}\beta(P(x,t),x,t,a)i(x,t,a)dar(x,t,0)=\epsilon_{r}\int_{0}^{\infty}\beta(P(x,t),x,t,a)r(x,t,a)da\end{array}$ (1.6)

$\epsilon_{i}$ (resp. $\epsilon_{r}$) being the constant probability that the disease (resp. the immunization) be

vertically transmitted. One also requires no flux boundary conditions

$\partial_{\eta}s=\partial_{\eta}i=\partial_{\eta}r=0$ on the boundary of$\Omega$

.

(1.7)

Defining Problem (G.P.) as the set of equations $(1.1)-(1.4),we$ _{shall first look at the}

behavior as $tarrow\infty$ of solution to (G.P.) starting from a given initial condition _{$u(x, 0, a)$},

under specific assumptions designed to yield explicit thresholds. Next we briefly sketch

some results for the solutions to Problem (S.I.R.S.), defined as the set of relations $(1.1)-$

(1.7). A special effort is made for the (S.I.S) model i.e. $f_{r}=0$ and $\sigma=0$, in the case

of a stable and non trivial T-time periodic or stationary endemicstate, generatedby the

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Main assumptions and

notations.

The diffusioncoefficent$k$is positive; $\Omega$is a bounded domain in$1R^{d}$with nice boundary.

Any function introduced in our models

is

nonnegative and smooth enough. In order

to have a simple description of the large time behaviour of the solutions one assumes :

(H1) $\{\begin{array}{l}\mu(p,x,t,a)=\mu_{n}(a)+\mu_{e}(p,x),\beta(p,x,t,a)=\beta_{n}(a)\beta_{e}(p,x)\sup p\mu_{n}compact,\sup p\beta_{n}\subset[0,A_{1}],A_{1}=\max\sup p\beta_{n}\end{array}$

Herein, $\mu_{n}$ (resp. $\beta_{n}$) is the natural death-rate (resp. birth-rate) while $\mu_{e}$ and$\beta_{e}$ take care

of spatial heterogeneities and density dependence, yielding a logistic effect. The external

supplies $f,$$f_{s},$$f_{l}$ and $f_{r}$ are such that $f=f_{s}+f_{i}+f_{r}$ and

$\{\begin{array}{l}0\leq f_{s}(x,t,a),f_{i}(x,t,a),f_{r}(x,t,a)\leq f(x,t,a)\leq m<\infty 0\leq F(x,t)=\int_{0}^{\infty}f(x,t,a)da\leq M<\infty\end{array}$

For the epidemic model, one also asks $\delta,$$\sigma$ and $\rho$ to depend only on the age variable, and

$supp\delta,$ _{$supp\sigma,$} _$supp\rho$ to be compact, while $\gamma$ is eitherindependent of time: $\gamma(t, a)=$

$\gamma(a)$ or time periodic : there is $T>0$ such that _{$\gamma(t+T, a)=\gamma(t, a)$}, and $supp\gamma_{0}\cup$

$supp\gamma_{1}\subset[0,$$\infty[\cross[0, A_{2}],$ $A_{2}<+\infty$.

In order to have non trivial solutions above the characteristic line $t=a$, at least when

$f=0$ on $\Omega\cross[0,$ $\infty[\cross[0, A_{1}]$, the initial distributions of individuals at time $t=0$ are

assumed to be fertile, moreprecisely

$suppu(., 0, .)\cap\Omega\cross[0, A_{1}]$ non empty.

We shall use the notation

$\psi_{*}(a)=\inf\{\psi(p, x,t, a),p, x,t\};\psi^{*}(a)=\sup\{\psi(p, x,t, a),p,x,t\}$

2 _{SINGLE SPECIES POPULATION DYNAMICS}

Inthis sectionwe analyze the largetime behaviour ofsolutionsto Problem (G.P.). Specific

notations are needed:

$\bullet$ $r$ is the root of the characteristic equation

$1= \int_{0}^{\infty}\beta_{n}(a)\pi(a)e^{-ra}da$, $\pi(a)=\exp(-\int_{0}^{a}\mu_{n}(\alpha)d\alpha)$ ;

$\bullet$ $\lambda_{1}$ is the dominant eigenvalue

$of-k\Delta+\mu_{e}(x)$in$\Omega$with Nemann boundary conditions

and $w_{1}$ is an associated positive eigenfunction (this makes sense when $\mu_{e}$ does not

depend on the variable$p$).

We begin with the linear case.

Theorem 1 Assume $\beta_{e}=1,$$\mu_{e}(p, x)=\mu_{e}(x)$ and let $f=0$

.

Then any solution to

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- when $r>\lambda_{1}$, $u(x,t, a)$ $arrow$ $+\infty$ (exponentially),

- when $r<\lambda_{1}$, $u(x, t, a)$ $arrow$ $0$ (exponentially),

- when $r=\lambda_{1},$ $u(x,t,a)$ $arrow c\pi(a)w_{1}(x),$ $c=c(u(x,0, a))>0$,

the convergence being

_uniform

on $\Omega\cross[0, A]$

for

each $A>0$.

The proofis quitesimilarto that in Langlais (1988), usingaseries expansion of$u$ over

the eigenfunctions of the diffusion operator.

When $r<\lambda_{1}$ a natural question to be asked is : can we transform the exponential

decay into a stabilisation toward anon trivial state upon supplying a non trivial input of

individiduals ? In the periodical case one finds a positive answer.

Theorem 2 Assume $\beta_{e}=1,$$\mu_{e}(p, x)=\mu_{e}(x)$ and _$leif$ be a nonnegative, bounded and

time periodic, with period$T>0_{f}$ function, $f(x, t, a)\not\equiv 0$ on $\overline{\Omega}\cross[0, T]\cross[0, A_{1}]$. Then

$\bullet$ - when $r\geq\lambda_{1}$ any solution to Problem $(G.P.)$ goes $to+\infty$ as $tarrow+\infty$ (in the way

defined

in Theorem 1)

$\bullet$ - when $r<\lambda_{1}$ there exists a unique non negative time periodic, with period $T_{f}$

solution to Problem $(G.P.)$ and it is globally stable in the class

ofbounded

solutions.

A proof is found in Kubo and Langlais (1991). When $r>\lambda_{1}$ it follows from a

com-parison argument and Theorem 1. When $r<\lambda_{1}$ the existence part uses a monotone

approximation process starting from a supersolution, while uniqueness and stability are

straightforward consequences of Theorem 1. Lastly the case $r=\lambda_{1}$ requires a specific

calculation.

In the nonlinear setting, things get more involved, even in the autonomons case with

no external supply $(f=0)$ and no spatial structure involved $(d=0)$ : an example given

by Swick (1981) shows that a non trivial periodic solution to (G.P.) may exist when $\beta$

depend, on both $p$ and $a$ but not as in (H1), whileBusenberg and Iannelli (1985) proved

that, when (H1) holds and $\beta_{e}=1$ , no non trivial periodic solution can exist. When no

external input of individuals is supplied, an analysis of the stabilization of solutions to

(GP) toward stationary states is performed in Langlais (1988) which we summarizenow.

By a stationary state we mean a solution to

(SGP) $\{\begin{array}{l}\partial_{a}v-k\triangle_{x}v+\mu(Q(x),x,a)v=g(x,a)x\in\Omega,a>Ov(x,0)=\int_{0}^{\infty}\beta(Q(x),x,a)v(x,a)da,x\in\Omega\partial_{\eta}v=0,x\in\partial\Omega,a>0Q(x)=\int_{0}^{\infty}v(x,a)da,x\in\Omega\end{array}$

The main two ingredients are: an a priori bound for $P(x, t)$ and some monotone

depen-dence of$\mu$ and $\beta$ on the variable_$p\ldots$ when they do not depend on the variable$a$ ! Under

assumption(Hl) this now reads

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(H2) $parrow\mu_{e}(p, .)$ non decreasing ;$parrow\beta_{e}(p, .)$ non increasing,$\beta_{e}(0, x)=1$.

It is then possible to define a suitable $\omega$-limit set for $\{u(x, t, a), t>0\}$ and to prove that

each element in it is a nonnegative stationary solution.

Two typical consequences concerning the stability of the trivial stationary state are

Theorem 3 Assume $(H2)$ hold and let _$f=0$. Let $\lambda_{10}$ be the dominant eigenvalue in $\Omega$

with Neumann boundary conditions value $of-k\triangle_{x}+\mu_{e}(0, x)$ Then

$\bullet$ - when $r\geq\lambda_{10}$ the trivial stationary state is not stable in the class

of

solutions

of

$(G.P.)$ having a non trivial initial condition at $t=0$.

$\bullet$ -when $r<\lambda_{10}$ the trivial stationary state is stable in the class

of

solutions

of

$(GP)$

having a non trivial initial condition at $t=0$.

The structure of the solution set for (SGP) is not known in general. We now give

two examples for which the existence and uniqueness of a positive and stable stationary

solution can be derived.

Example 1 Assume $\beta(p, x, a)=\beta_{n}(a)$ ; then one may show that anystationary solution

is separable, i.e. $v(x, a)=\varphi(a)Q(x)$ where

$-k\triangle Q=(r-\mu_{e}(Q, x))Q$ in $\Omega$, _{$\partial_{\eta}Q=0$} on $\partial\Omega$,

$r$ being the root of the above characteristic equation, while

$\varphi’+(\mu_{n}(a)+r)\varphi=0$ in $a>0, \varphi(0)=\int_{0}^{\infty}\beta_{n}(a)\varphi(a)da,$ $\int_{0}^{\infty}\varphi(a)da=1$

The monotonicity property requiredin (H2) for $\mu_{e}$ impliesthat thereis at most one

non trivial and nonnegative stationary solution ; furthermore it is stable from the

first part of Theorem 3 as soon as it exists.

Example 2 Assumenow that $\mu_{n}=0$ and $\beta(p, x, a)=\beta_{e}(p, x)$ ; then given any

nonnega-tive and non trivial solution to (SGP) one may check upon integrating overall ages

that $Q$ is a nonnegative solution to

$-k\triangle Q=(\beta_{e}(Q, x)-\mu_{e}(Q, x))Q$ in $\Omega$,

$\partial_{\eta}Q=0$ on $\partial\Omega$

.

Again under condition (H2) there is at most one non trivial and nonnegative

sta-tionary solution and it is stable from the first part of Theorem 3.

Assuming a periodic input of individuals is supplied, an analysis of periodic solutions

to problem (GP) is made in Kubo and Langlais (to appear) ; we give two simple results

from it to which we refer for a more comprehensive treatment.

Theorem 4 Assume $(H2)$ holds; let $f$ be a time periodic, with period$T$, and nonnegative

function, $f(x, t, a)\not\equiv 0$ on$\overline{\Omega}\cross[0, T]\cross[0, A_{1}]$, and let $\lambda_{10}$ be as is Theorem 3. Then

$\bullet$ when $r\geq\lambda_{10}$, any solution to $(GP)$ tends $to+\infty$ as $tarrow+\infty$

$\bullet$ - when $r<\lambda_{10}$, there is at least a nonnegative, T-time periodic and nonnegative

solution to $(GP)$

.

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3 THE S.I.R.S.

MODEL

Little is known for the complete S.I.R.S. model. Most of available results are derived

when $f=0$ and no spatial structure involved ; they concem the stabilization toward

stationary solutions and the stability of the trivial endemic state. A partial uniqueness

result is given in Inaba (1989) while the method in Lafaye and Langlais (manuscript)

carries over to spatially structured populations.

The existence oftime periodic solutions for the complete S.I.R.S. model is analyzed

in Kubo and Langlais (to appear).

To conclude this short section let us say that the structure of the stationary solutions

set (when $f=0$) and of the T-time periodic solution set (for T-time periodic data) for

Problem (S.I.R.S.) is not known.

4 THE S.I.S. MODEL WITH VERTICAL

TRANS-MISSION

The S.I.S. model corresponds to the case wherein $f_{r}=0$ and $\sigma=0$ implying that the

removed class is empty: $r=0$. One now has

$s=u-i$

so that the model reduces to

(S.I.S.) $\{\begin{array}{l}Di+[\mu(P(x,t),x,a)+\delta(a)]i=\gamma(t,a)\varphi(i)[u(x,t,a)-i]+f_{i}(x,t,a)i(x,t,0)=\epsilon_{i}\int_{0}^{\infty}\beta(P(x,t),x,a).i(x,t,a)da\partial_{\eta}i(x,t,a)=_{l}0ontheboundaryof\Omega\end{array}$

In this setting, interesting results concerning the uniqueness and stability of non trivial

stationary or time periodic solutions canbe derived. Most results of this section are taken

from Busenberg and Langlais (in preparation).

Theorem 5 Let $\gamma$ and $f_{i}$ be nonnegative and T-time periodic

functions

; assume that

there is a nonnegative T-time periodic solution $u$

of

Problem $(G.P.)$ non trivial on $\overline{\Omega}\cross$

$[0, T]\cross[0, A_{1}]$

.

Then there is a maximal T-time periodic solution

of

Problem (S.I.$S$) in

the range $0\leq j\leq u$

.

The proof uses ideas in Langlais (manuscript) for the intracohort model whithout

diffusion. Actually, a suitable modification of the partial differential equation yields

the following : the semi-orbit $\{i(x,t,a),t>0\}$ corresponding to the initial condition

$i(x, 0, a)=u(x, 0, a)$ is such that $0\leq i(x,t+(n+1)T,$$a$) $\leq i(x,t+nT, a)\leq u(x,t, a)$,

for any $n\geq 0$

.

In the limnit $narrow+\infty$ one has a nonnegative T-periodic solution. It is the

maximal solution in the desired range from a comparison principle.

When $f_{i}(x, t, a)\not\equiv 0$ on $\overline{\Omega}\cross[0, T]\cross[0, \infty$), it is quite clear that this maximal solution

is not the trivial one. Otherwise, when$f_{i}=0$ on$\overline{\Omega}\cross[0, T]\cross[0, A_{1}]$, this maximal solution

is not the trivial one on that domain if one can find a non trivial and nonnegative

sub-solution (as it is easily done when no spatial structure is involved) or if this assumption

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Nowregardless ofthese sufficient conditions to get a nontrivialsolution, a uniqueness

and stability result may be proved, provided some a priori positivity result be granted.

Set (see Busenberg at all (manuscript))

(H3) $\{\sigma_{- o^{1}rf_{i}^{i}=0ande1se\gamma(t,a)\not\equiv 0on[0}^{- eitherf_{i}(x_{1},t,a)\not\equiv 0_{0}on\overline{\Omega}\cross[0,T]}- orf=0andthereexist\sigma>0an_{n^{d}t_{T]\cross[0,A]inthe}^{\gamma(a)^{]}\not\equiv 0_{1}on[0,A]suchthat}}^{\cross[0_{11}A}\gamma 1(a)\leq\gamma(t,a)\leq\gamma_{11}(a),a>0,ihein^{1}tercohortcase_{intracohort}^{1}$

case.

Theorem 6 Let $\gamma,$$f,$$f_{i}$ and $u$ be as in Theorem 5. Assume $0<\epsilon_{1}\leq 1$ and $(H3)$

holds. Then there is at most one T-time periodic solution to Problem (S.I.$S$) in the

range $0\leq j\leq u$ and non trivial on St $\cross[0, T]\cross[0, +\infty$). Furthermore it is globally stable

in the range

_of

solutions to Problem (S.I.$S.$) having a

fertile

initial condition, such that

$0\leq i(x, 0, a)\leq u(x, 0, a)$

.

The proof is derived upon adapting techniques developped in Busenberg et al

(manuscript) to dealwith stationarysolutions for ageneralforce of infection termwithout

spatial structure. Under assumption (H3) one may show that, for theverticaltransmission

case, any T-time periodic solution of (S.I.S.),non trivial on$\overline{\Omega}\cross[0, T]\cross[0, A_{1}]$ is actually

positive on this domain. Uniqueness and stability follow from a concavity argument.

We now apply our results toanswerthree questions of relevantepidemiologicalinterest.

Question 1 Assuming no external supply of infected individuals, can an initial input

of infected individuals generate a stationary and stable non trivial endemic state

within a global population at stationary state ?

The setting is $f_{i}=0,$$u(x,t, a)=v(x, a)$ and $P(x,t)=Q(x)$ a non trivial solution

to Problem (S.G.P.), while$\gamma_{0}(t, a)=\gamma_{0}(a)$ and $\gamma_{1}(t, a)=\gamma_{1}(a)$. From Theorem 6,

one has at most one non trivial stationary solution ifeither $\gamma_{1}(a)\not\equiv 0$ or $\gamma_{0}(a)\not\equiv 0$

on $[0, A_{1}]$, and it is stable in a suitable range. Henceone are left with finding a non

trivial solution. Let us consider the intracohort case. Going back to the sketched

proof of Theorem 5, the maximal stationary solution is obtained as the decreasing

limit as $tarrow+\infty$ of _{$\{i(x,t, a), t>0\}$} provided $i(x, 0, a)=v(x, a)$ ; arguing as in

Lafaye and Langlais (1993) one may prove that this convergence is uniform on any

compact domain $\overline{\Omega}\cross[0, A],$ $A>0$

.

Given any small positive $\alpha$, if this maximal

solution is the trivial one there exists $T(\alpha)>0$ such that

$0\leq i(x,t, a)\leq\alpha,t\geq T(\alpha),$$x\in\Omega,$_{$0 \leq a\leq\max supp\gamma_{0}$}.

From a comparison principle one may show $0\leq w(x,t, a)\leq i(x, t, a),$ $w$ being the

solution to the linear problem in $\Omega\cross[T(\alpha),$$\infty$) $\cross[0, \infty$) :

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and such that $\omega(x, T(\alpha),$$a$) $=i(x, T(\alpha),$$a$). Setting $\lambda_{11}$ the dominant eigenvalue

$of-k\triangle+\mu_{e}(P(x), x)$ in $\Omega$ with Neumann boundary conditions and

$r_{*}$ the root of

the characteristic equation (see section 2) with $\beta_{\eta}(a)=\epsilon_{i}\beta_{*}(a)$ and

$\mu_{\eta}$ replaced by

$\mu_{\eta}+\delta-\gamma_{0}v_{*}$, iffollows from Theorem 1, that $r_{*}>\lambda_{11}$ implies $\omega(., t, .)arrow+\infty$ as

$tarrow+\infty$ a contradiction to the maximalsolution beingthe trivial one. Hence when

$r_{*}>\lambda_{11}$ the answer ispositive; converselyif$r^{*}<\lambda_{11}$ (withobvious notations), one

may show that the maximal solution is the trivial one and the answer is negative.

Question 2 Assuming no external supply of infected individuals, can a T-time periodic

force of infection generate a stable and non trivial T-time periodic endemic state

within a population at stationary state ?

The setting is as in Question 1 for $f_{i}$ and $(u, P)$ but now $\gamma$ is a T-time periodic

function. Again, Theorem 6 yields at most one nontrivial T-periodicendemicstate

when $(H3)$ holds and it is stable. A positive subsolution may be constructed upon

.using a non trivial stationary solution for Problem (S.I.S), the contact rate being

givenby either$\gamma_{0*}$ or$\gamma_{1*}$ : in that case the answer is positive. Conservely if there is

no non trivial stationary solutions when the contatc rate is given by $\gamma_{0}^{*}$ or $\gamma_{1}^{*}$, then

the answer is negative.

Question 3 Can a T-time periodic supply of infected individuals generatea T-time

pe-riodic and stable non trivialendemic state within a T-time periodic (or stationary)

global population ?

The setting is now $f$ T-time periodic (resp. time independent), $(u, P)$ a T-time

periodic solution of (G.P.) (resp. non trivial solution to (S.G.P.)) and $f_{i}$ a T-time

periodic function. From Theorem 5 and 6 it follows that the answer is positive as

soon as $f_{i}(x, t, a)\not\equiv O$ on $\overline{\Omega}\cross[0, T]\cross[0, A_{1}]$, regardless of the contact rate term.

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This work is partially supported by the CNRS, Programme Environnement,